364 APPENDIX: ELEMENTS OF LINEAR ALGEBRA
(A.3.2) Now let E be a vector space, let (Ma)a 61 be a family of subspaces of
E, and for each a e I let /a : Ma ~»E be the canonical injection. By (A.3.1)
there exists a linear mapping /of M = © Ma into E such that
ael
/0 ja=/a for all ael. The image of/is the subspace M' = % Ma of E.
ael
If/is injective, the sum £ Ma of subspaces of E is said to be direct. This
ael
therefore means that every vector x belonging to % Ma can be expressed
ael
uniquely in the form £ xa, where H is a finite subset of I, and xx e Ea and
aeH
xa 7* 0 for all a e H. (If x = 0 we have to take H = 0.)
We shall usually identify £ Ma and © Ma by means off.
ael ael
(A.3.3) The sum of a family (Ma)aei of subspaces of a vector space is direct
if and only if Ma n £ M^ = {0} for all ael.
0*a
For to say that the mapping / defined in (A.3.2) is not injective means
that there exists a finite number of nonzero elements xXi e Mai (1 ^ / < n)
such that £ xai = 0 (A.2.2). This equation can be written in the form
x*i = H (~*«A an(* expresses that xai belongs to Mai n £ MA. Hence the
j>l J
result.
(A.3.4) If E is equal to the direct sum of two subspaces M and N, we say
that M and N are supplementary subspaces of E (or that M (resp. N) is a sup-
plement to N (resp. M)). This signifies that M + N = E and M n N = {0}.
Hence we have linear mappings/?: E -> M and q : E -> N, the " projections"
of E onto M and N, such that x = p(x) + q(x) for all x e E. The kernel of p
(resp. q) is N (resp. M), and p(p(x)) = p(x), q(q(x)) = q(x).
We cannot speak of "the" supplementary subspace of a subspace M,
because in general there is more than one. However, there is the following
result:
(A.3.5) Let M, N, N' be subspaces of E, such that M and N are supplementary,
and M and N' are supplementary. Let q : E ->• N be the projection of E onto
N corresponding to the direct sum decomposition E = M © N. Then the
restriction q': x-*q(x) ofq to N' is an isomorphism 0/N' onto N.